Equal wavenumber

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Equal wavenumber

The case can be reached most easily by setting and $k_b = k + \delta$ while taking the limit $\delta \rightarrow 0$ . The solution now has to be spatially scaled (dilated about the arbitrary origin) appropriately for its new wavenumber. Section 6.1.1 (also Appendix I) gives the expansion of the dilation operator at wavenumber in powers of $\delta$ , thus

$\begin{displaymath} b(k + \delta;{\mathbf r}) \; = \; \left[ 1 + \frac{\delta}{... ...} \cdot \nabla + O(\delta^2) \cdots \right] b(k;{\mathbf r}) . \end{displaymath}$

(H.6)

Substitution into (H.5), noting $k_a^2 - k_b^2 = -2 \delta k + O(\delta^2)$ , and taking the limit (in which only the lowest order $O(\delta^0)$ survives) gives

$\begin{displaymath} \left\langle a \left\vert b\right.\right\rangle _{\mathcal{... ...nabla ({\mathbf r}\cdot\nabla b) \rule{0in}{0.2in} \right] . \end{displaymath}$

(H.7)

This can be shown to be a generalization of the overlap formula given by Boasman [33] to general

and differing solutions

and

. Despite the derivation using scaling, (H.7) is in fact an identity, which can be proved using a messy algebra sequence^H.1. Unlike the LHS, the RHS is not manifestly $a^*\!{\leftrightarrow}b$ symmetric. However, it can be written in the symmetrized form (H.14) derived in the next section. Notice that the choice

has increased the order of derivative required on the boundary by one.

In the case of Dirichlet BCs the second term in (H.7) vanishes, and the replacement ${\mathbf r}\cdot\nabla b = ({\mathbf r}\cdot{\mathbf n}) {\mathbf n}\cdot\nabla b$ (which follows from the fact that $\nabla b$ and ${\mathbf n}$ are parallel), gives

$\begin{displaymath} \left\langle a \left\vert b\right.\right\rangle _{\mathcal{... ...l_n a^* \partial_n b , \hspace{0.5in} \mbox{(Dirichlet BCs)}. \end{displaymath}$

(H.8)

However the knowledge that the LHS is zero for $b \ne a$ (orthogonal eigenstates) shows that (H.8) is an exact boundary orthogonality relation for degenerate Dirichlet eigenstates. Note that the RHS is proportional to the matrix element $(\partial {\mathcal{H}} / \partial x)_{ab}$ of the billiard dilation deformation between degenerate states (Appendix C). On the other hand, choosing

we have

$\begin{displaymath} \left\langle a \left\vert a\right.\right\rangle _{\mathcal{... ...\partial_n a \vert^2 , \hspace{0.5in} \mbox{(Dirichlet BCs)}. \end{displaymath}$

(H.9)

This very useful boundary formula for the norm of Dirichlet eigenstates appears to have been found first for

by Berry and Wilkinson (appendix of [28]). It was since derived in a different way by Boasman[33], and for general

was derived in our work[14]. However, I believe the new derivation above to be the simplest yet.

Next: Matrix trick for pushing Up: Overlaps: Previous: Differing wavenumbers

Alex Barnett 2001-10-03